M 120X[ G) B2](C+NI ONU)) spanned the l- space N56 355 spanned the l 3 Theorem 2 Equality of Row and Column Ranh The column rank and row rank of a matric are equal By the definition of row rank and its co unterpart for column rank the row space and column space of a matric have the same dimension Theorem 3 rank bab. min bran k ba. rank bB Theorem 4 For any matric a and nonsingular matrices B and C, the rank af bac is equal to the rank of a (The meaning of nonsingular matrices will be introduced later) Theorem 5 rank ba. t rank ba'A Definition 5 Determinant of a matric For an n matric(square matria), the area of the matric is the determinant det AT JA|x4×3-U×N Proposition 1 The determinant of a matric is nonzero if and only if it has full ranh rank ba. r dim ba Definition 6 Inverse of a matriT Suppo se that we could find a square matrit B such that Ba x I, B is the inverse of A, denoted Br A-I Example 8 n 3 Definition 7 Nonsingular Matric A matrit who se inverse exists is nonsingularMATRIX ALGEBRA (CONTINUE) 3 spanned the R4 space C =   1 5 6 3 2 1 4 1 3 5 5 4   spanned the R 3 space Theorem 2 Equality of Row and Column Rank The column rank and row rank of a matrix are equal. By the definition of row rank and its counterpart for column rank, the row space and column space of a matrix have the same dimension. Theorem 3 rank (AB) ≤ min (rank (A), rank (B)) Theorem 4 For any matrix A and nonsingular matrices B and C, the rank of BAC is equal to the rank of A. (The meaning of nonsingular matrices will be introduced later). Theorem 5 rank (A) = rank (A′A) Definition 5 Determinant of a matrix For a n × n matrix (square matrix), the area of the matrix is the determinant. Example 7 A =
4 2 1 3 det A = |A| = 4 × 3 − 2 × 1 = 10 Proposition 1 The determinant of a matrix is nonzero if and only if it has full rank. rank (A) = dim (A) Definition 6 Inverse of a matrix Suppose that we could find a square matrix B such that BA = I, B is the inverse of A, denoted B = A−1 Example 8 A =
4 2 1 3 B = A−1 =
3 10 − 1 5 − 1 10 2 5 Definition 7 Nonsingular Matrix A matrix whose inverse exists is nonsingular.